Invertible Matrix - Properties

Properties

Let A be a square n by n matrix over a field K (for example the field R of real numbers). The following statements are equivalent:

A is invertible.
A is row-equivalent to the n-by-n identity matrix In.
A is column-equivalent to the n-by-n identity matrix In.
A has n pivot positions.
det A ≠ 0. In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring.
rank A = n.
The equation Ax = 0 has only the trivial solution x = 0 (i.e., Null A = {0})
The equation Ax = b has exactly one solution for each b in Kn.
The columns of A are linearly independent.
The columns of A span Kn (i.e. Col A = Kn).
The columns of A form a basis of Kn.
The linear transformation mapping x to Ax is a bijection from Kn to Kn.
There is an n by n matrix B such that AB = In = BA.
The transpose AT is an invertible matrix (hence rows of A are linearly independent, span Kn, and form a basis of Kn).
The number 0 is not an eigenvalue of A.
The matrix A can be expressed as a finite product of elementary matrices.

Furthermore, the following properties hold for an invertible matrix A:

  • (A−1)−1 = A;
  • (kA)−1 = k−1A−1 for nonzero scalar k;
  • (AT)−1 = (A−1)T;
  • For any invertible n-by-n matrices A and B, (AB)−1 = B−1A−1. More generally, if A1,...,Ak are invertible n-by-n matrices, then (A1A2Ak−1Ak)−1 = Ak−1Ak−1−1⋯A2−1A1−1;
  • det(A−1) = det(A)−1.

A matrix that is its own inverse, i.e. A = A-1 and A2 = I, is called an involution.

Read more about this topic:  Invertible Matrix

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